DNMW04 - New trends and challenges in the mathematics of optimal design

In this talk I will discuss the question of the maximization of the $k$-th eigenvalue of the Neumann-Laplacian under a volume constraint. After an introduction to the topic I will discuss the existence of optimal geometries. For now, there is no a general existence result, but one can prove existence of an optimal {\it (over) relaxed domain}, view as a density function. In the second part of the talk, I will focus on the low eigenvalues. The first non-trivial one is maximized by the ball, the result being due to Szego and Weinberger in the fifties. Concerning the second non-trivial eigenvalue, Girouard, Nadirashvili and Polterovich proved that the supremum in the family of planar simply connected domains of $R^2$ is attained by the union of two disjoint, equal discs. I will show that a similar statement holds in any dimension and without topological restrictions.